Finite Group Databasebeta

← #14960778

Subgroups — #14960778

Order 2816.

2450 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623014C211yesmaximal, characteristic, complemented
2#13076903C212yesmaximal, complemented
3#13075031C218yesmaximal, complemented
4#13074953C218yesmaximal, complemented
5#13074429C212yesmaximal, complemented
6#13073774C212yesmaximal, complemented
7#13073090C212yesmaximal, complemented
8#13072894C212yesmaximal, complemented
9#13071939C212yesmaximal
10#13071214C211yesmaximal, characteristic, complemented
11C22×C11×#123C211yesmaximal, characteristic
12C23×D4×C11C2211yescharacteristic
13C24×C4×C11C2211yescharacteristic
14C2×C11×#123—28no—
15C2×C11×#123C2218yes—
16C22×C11×C4⋊C4C2211yescharacteristic
17C22×C11×C4⋊C4C2211yescharacteristic
18C22×C11×C22⋊C4C2211yescharacteristic
19C22×C11×C22⋊C4C2211yescharacteristic
20C22×C11×C22⋊C4C2211yescharacteristic
21C22×C22⋊Dic11C2212yes—
22C2×#66988C2214yes—
23C2×#66977C2212yes—
24C2×#66977C2212yes—
25C2×#66977C2214yescomplemented
26#10623014—28no—
27C2×#66931C2214yescomplemented
28C2×#66931C2214yescomplemented
29#10622860C2212yes—
30#10622860C2214yes—
31#10622835C2218yes—
32#10622833C2214yes—
33#10622825C2212yes—
34#10622825C2214yescomplemented
35#10622824C2214yes—
36#10622823C2214yes—
37#10622823C2214yes—
38#10622823C2218yescomplemented
39C4×C22⋊Dic11C2212yes—
40#10622599C2214yescomplemented
41#10622597C2218yescomplemented
42#10622593C2218yescomplemented
43#10622592C2214yescomplemented
44#10622589C2214yescomplemented
45#10622587C2214yescomplemented
46#10622566C2214yescomplemented
47#10622565C2218yescomplemented
48#10622564C2218yescomplemented
49#10622560C2214yescomplemented
50#10622559C2214yescomplemented
51#10622558C2218yescomplemented
52#10622556C2214yescomplemented
53C25×C11C2311yescharacteristic
54C25×C11C2311yescharacteristic
55C22×D4×C11—24no—
56C22×D4×C11—44no—
57C22×D4×C11C2314yes—
58C23×C4×C11—24no—
59C23×C4×C11—44no—
60C23×C4×C11C2311yescharacteristic
61C23×C4×C11C2314yes—
62C23×C4×C11C2311yescharacteristic
63C23×C4×C11C2311yescharacteristic
64C23×C4×C11C2311yescharacteristic
65C23×C4×C11C2311yescharacteristic
66C23×Dic11C2312yes—
67C23×Dic11C2312yes—
68C22×Dic22C2312yes—
69C22×Dic22C2312yes—
70C11×#123—232no—
71C2×C11×C4⋊C4—24no—
72C2×C11×C4⋊C4—24no—
73C2×C11×C4⋊C4C2314yes—
74C2×C11×C4⋊C4C2314yes—
75C2×C11×C22⋊C4—24no—
76C2×C11×C22⋊C4—24no—
77C2×C11×C22⋊C4D418yes—
78C2×C11×C22⋊C4C2314yes—
79C2×C11×C22⋊C4C2314yes—
80C2×C11×C22⋊C4C2314yes—
81C2×C22⋊Dic11—28no—
82C2×C22⋊Dic11C2314yes—
83C2×C22⋊Dic11C2314yes—
84C2×C22⋊Dic11C2318yes—
85C2×C22⋊Dic11C2314yes—
86C2×C22⋊Dic11C2314yes—
87C2×C22⋊Dic11C2318yes—
88C2×C22⋊Dic11C2314yes—
89#66988—48no—
90C2×C4⋊Dic11C2312yes—
91C2×C4⋊Dic11C2312yes—
92C2×C4⋊Dic11C2314yes—
93#66977—28no—
94#66977—28no—
95#66977—48no—
96C2×Dic11⋊C4C2314yes—
97C2×Dic11⋊C4C2314yes—
98C2×Dic11⋊C4C2318yes—
99C2×Dic11⋊C4C2318yes—
100C2×C4×Dic11C2314yes—
101#66931—216no—
102#66931—216no—
103C11⋊#95C2318yes—
104C11⋊#95C2318yes—
105C11⋊#95C2318yes—
106C11⋊#95C2314yes—
107C11⋊#95C2314yes—
108C11⋊#95C2318yes—
109#36720—111nomaximal, Sylow
110C24×C11—22no—
111C24×C11—24no—
112C24×C11—24no—
113C24×C11—24no—
114C24×C11—42no—
115C24×C11—42no—
116C24×C11—42no—
117C24×C11C2×D414yes—
118C24×C11C2412yes—
119C24×C11C2412yes—
120C24×C11C2411yescharacteristic
121C2×D4×C11—28no—
122C2×D4×C11—28no—
123C2×D4×C11—44no—
124C2×D4×C11—416no—
125C22×C4×C11—22no—
126C22×C4×C11—22no—
127C22×C4×C11—22no—
128C22×C4×C11—28no—
129C22×C4×C11—24no—
130C22×C4×C11—28no—
131C22×C4×C11—42no—
132C22×C4×C11—44no—
133C22×C4×C11—42no—
134C22×C4×C11—42no—
135C22×C4×C11—42no—
136C22×C4×C11—416no—
137C22×C4×C11C2×D414yes—
138C22×C4×C11C2×D414yes—
139C22×C4×C11C2412yes—
140C22×C4×C11C2412yes—
141C22×C4×C11C2412yes—
142C22×C4×C11C2412yes—
143C22×C4×C11C2412yes—
144C22×Dic11—24no—
145C22×Dic11—24no—
146C22×Dic11—44no—
147C22×Dic11—44no—
148C22×Dic11C2418yes—
149C22×Dic11C2414yes—
150C22×Dic11C2414yes—
151C2×Dic22—24no—
152C2×Dic22—24no—
153C2×Dic22—44no—
154C2×Dic22—44no—
155C11×C4⋊C4—28no—
156C11×C4⋊C4—28no—
157C11×C22⋊C4—28no—
158C11×C22⋊C4—28no—
159C11×C22⋊C4C2×D4116yes—
160C22⋊Dic11—28no—
161C22⋊Dic11—28no—
162C22⋊Dic11—216no—
163C22⋊Dic11—28no—
164C22⋊Dic11—44no—
165C22⋊Dic11—48no—
166C22⋊Dic11—44no—
167C4⋊Dic11—24no—
168C4⋊Dic11—24no—
169C4⋊Dic11—44no—
170Dic11⋊C4—28no—
171Dic11⋊C4—28no—
172Dic11⋊C4—48no—
173Dic11⋊C4—48no—
174C4×Dic11—44no—
175C2×#543—111no—
176C22×#123D1111yescharacteristic
177#2848—112no—
178#2687—118no—
179#2680—118no—
180#2627—112no—
181#2561—112no—
182#2502—112no—
183#2483—112no—
184#2405—112no—
185#2312—111no—
186C23×C11—22no—
187C23×C11—22no—
188C23×C11—28no—
189C23×C11—22no—
190C23×C11—24no—
191C23×C11—22no—
192C23×C11—24no—
193C23×C11—24no—
194C23×C11—24no—
195C23×C11—24no—
196C23×C11—28no—
197C23×C11—44no—
198C23×C11—44no—
199C23×C11—416no—
200C23×C11—44no—
201C23×C11—44no—
202C23×C11—88no—
203C23×C11C22×D414yes—
204C23×C11C22×D412yes—
205C23×C11H(5,𝔽2)14yes—
206C23×C112−1+414yes—
207C23×C11C2511yescharacteristic
208D4×C11—416no—
209C2×C4×C11—24no—
210C2×C4×C11—24no—
211C2×C4×C11—22no—
212C2×C4×C11—22no—
213C2×C4×C11—22no—
214C2×C4×C11—24no—
215C2×C4×C11—24no—
216C2×C4×C11—24no—
217C2×C4×C11—44no—
218C2×C4×C11—44no—
219C2×C4×C11—416no—
220C2×C4×C11—44no—
221C2×C4×C11—44no—
222C2×C4×C11—44no—
223C2×C4×C11C22×D414yes—
224C2×C4×C11C22×D414yes—
225C2×Dic11—24no—
226C2×Dic11—24no—
227C2×Dic11—28no—
228C2×Dic11—28no—
229C2×Dic11—28no—
230C2×Dic11—28no—
231C2×Dic11—28no—
232C2×Dic11—84no—
233C2×Dic11—84no—
234Dic22—84no—
235Dic22—84no—
236C23×D4C2×D1111yescharacteristic
237C24×C4C2×D1111yescharacteristic
238#543—228no—
239C2×#123—28no—
240C2×#123—114no—
241C2×#123C2×D1118yes—
242C2×C22⋊Q8—112no—
243C2×C22⋊Q8—112no—
244C2×C22⋊Q8—114no—
245C2×C22⋊Q8—114no—
246C2×C22⋊Q8—114no—
247C22×C4⋊C4C2×D1111yescharacteristic
248C22×C4⋊C4C2×D1111yescharacteristic
249C22×C22⋊C4—112no—
250C22×C22⋊C4C2×D1111yescharacteristic
251C22×C22⋊C4C2×D1111yescharacteristic
252C22×C22⋊C4C2×D1111yescharacteristic
253#400—114no—
254#398—118no—
255#398—118no—
256#397—114no—
257#397—118no—
258#395—114no—
259#394—118no—
260#393—114no—
261#391—114no—
262#388—118no—
263#388—118no—
264#386—114no—
265#385—114no—
266#385—118no—
267#385—114no—
268#385—114no—
269#384—114no—
270#381—114no—
271#381—114no—
272#380—112no—
273#380—114no—
274#379—114no—
275#379—112no—
276C4×C22⋊C4—112no—
277C22×C11—22no—
278C22×C11—22no—
279C22×C11—28no—
280C22×C11—24no—
281C22×C11—24no—
282C22×C11—24no—
283C22×C11—24no—
284C22×C11—28no—
285C22×C11—416no—
286C22×C11—42no—
287C22×C11—42no—
288C22×C11—44no—
289C22×C11—48no—
290C22×C11—88no—
291C22×C11#53614yes—
292C22×C11C23×D411yescharacteristic
293C22×C11C2×H(5,𝔽2)12yes—
294C22×C11C2×2−1+412yes—
295C22×C11#58512yes—
296C4×C11—24no—
297C4×C11—24no—
298C4×C11—42no—
299C4×C11—42no—
300C4×C11—42no—
301Dic11—44no—
302Dic11—44no—
303Dic11—48no—
304C25C22×D1111yescharacteristic
305C25C22×D1111yescharacteristic
306C22×Q8—112no—
307C22×Q8—112no—
308C22×D4—24no—
309C22×D4—44no—
310C22×D4C22×D1114yes—
311C23×C4—24no—
312C23×C4—44no—
313C23×C4—112no—
314C23×C4—112no—
315C23×C4C22×D1111yescharacteristic
316C23×C4C22×D1114yes—
317C23×C4C22×D1111yescharacteristic
318C23×C4C22×D1111yescharacteristic
319C23×C4C22×D1111yescharacteristic
320C23×C4C22×D1111yescharacteristic
321#123—232no—
322#123—448no—
323C22⋊Q8—228no—
324C22⋊Q8—228no—
325C22⋊Q8—2216no—
326C22⋊Q8—2216no—
327C22⋊Q8—448no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—112no—
331C2×C4⋊C4—112no—
332C2×C4⋊C4—114no—
333C2×C4⋊C4—114no—
334C2×C4⋊C4—114no—
335C2×C4⋊C4—118no—
336C2×C4⋊C4—118no—
337C2×C4⋊C4C22×D1114yes—
338C2×C4⋊C4C22×D1114yes—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—114no—
342C2×C22⋊C4—114no—
343C2×C22⋊C4—118no—
344C2×C22⋊C4—114no—
345C2×C22⋊C4—114no—
346C2×C22⋊C4—118no—
347C2×C22⋊C4—114no—
348C2×C22⋊C4—228no—
349C2×C22⋊C4C22⋊D1118yes—
350C2×C22⋊C4C22×D1114yes—
351C2×C22⋊C4C22×D1114yes—
352C2×C22⋊C4C22×D1114yes—
353C2×C42—114no—
354#95—114no—
355#95—118no—
356#95—114no—
357#95—118no—
358#95—118no—
359#95—118no—
360C2×C11—24no—
361C2×C11—44no—
362C2×C11—44no—
363C2×C11C2×#53611yescharacteristic
364C2×C11#348411yescharacteristic
365C2×C11#348511yescharacteristic
366C2×C11#353514yes—
367C24—24no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24C2×C22⋊D1114yes—
375C24C23×D1112yes—
376C24C23×D1112yes—
377C24C23×D1111yescharacteristic
378C2×Q8—224no—
379C2×Q8—224no—
380C2×Q8—444no—
381C2×Q8—444no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—44no—
385C2×D4—416no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—416no—
398C22×C4—118no—
399C22×C4—114no—
400C22×C4—114no—
401C22×C4—224no—
402C22×C4—224no—
403C22×C4—444no—
404C22×C4—444no—
405C22×C4C2×C22⋊D1114yes—
406C22×C4C2×C22⋊D1114yes—
407C22×C4C23×D1112yes—
408C22×C4C23×D1112yes—
409C22×C4C23×D1112yes—
410C22×C4C23×D1112yes—
411C22×C4C23×D1112yes—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—228no—
415C4⋊C4—228no—
416C4⋊C4—224no—
417C4⋊C4—224no—
418C4⋊C4—448no—
419C4⋊C4—444no—
420C4⋊C4—448no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—2216no—
424C22⋊C4—228no—
425C22⋊C4—228no—
426C22⋊C4—228no—
427C22⋊C4—444no—
428C22⋊C4—444no—
429C22⋊C4—448no—
430C22⋊C4C2×C22⋊D11116yes—
431C42—444no—
432C11#3672011yescharacteristic, complemented, Sylow
433C23—22no—
434C23—22no—
435C23—28no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—28no—
444C23—44no—
445C23—44no—
446C23—416no—
447C23—44no—
448C23—44no—
449C23—88no—
450C23#6703714yes—
451C23#6704314yes—
452C23C22×C22⋊D1114yes—
453C23C22×C22⋊D1112yes—
454C23C24×D1111yescharacteristic
455Q8—884no—
456Q8—884no—
457D4—416no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—416no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—228no—
473C2×C4—228no—
474C2×C4—228no—
475C2×C4—228no—
476C2×C4—228no—
477C2×C4—224no—
478C2×C4—224no—
479C2×C4—884no—
480C2×C4—884no—
481C2×C4C22×C22⋊D1114yes—
482C2×C4C22×C22⋊D1114yes—
483C22—22no—
484C22—22no—
485C22—28no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—28no—
491C22—416no—
492C22—42no—
493C22—42no—
494C22—44no—
495C22—48no—
496C22—88no—
497C22#1062335614yes—
498C22C2×#6703712yes—
499C22C2×#6704312yes—
500C22#1062345312yes—
501C22C23×C22⋊D1111yescharacteristic
502C4—24no—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—444no—
508C4—444no—
509C4—448no—
510C2—24no—
511C2—44no—
512C2—44no—
513C2C2×#1062335611yescharacteristic
514C2#1308225111yescharacteristic
515C2#1308226011yescharacteristic
516C2#1308282914yes—
subgroupG/H
center, FrattiniC23C24×D11
commutator, socle, cosocleC23×C11C25
FittingC22×C11×#123C2
prime radicalC23×D4×C11C22
hypercenterC22×#123D11
hypocenterC11#36720
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×#123

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#10623014C2semidirect
2. #13076903C2semidirect
3. #13075031C2semidirect
4. #13074953C2semidirect
5. #13074429C2semidirect
6. #13073774C2semidirect
7. #13073090C2semidirect
8. #13072894C2semidirect
10. #13071214C2semidirect
25. C2×#66977C22semidirect
27. C2×#66931C22semidirect
28. C2×#66931C22semidirect
34. #10622825C22semidirect
38. #10622823C22semidirect
40. #10622599C22semidirect
41. #10622597C22semidirect
42. #10622593C22semidirect
43. #10622592C22semidirect
44. #10622589C22semidirect
45. #10622587C22semidirect
46. #10622566C22semidirect
47. #10622565C22semidirect
48. #10622564C22semidirect
49. #10622560C22semidirect
50. #10622559C22semidirect
51. #10622558C22semidirect
52. #10622556C22semidirect
432. C11#36720semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#106230142
#130769032
#130750312
#130749532
#130744292
#130737742
#130730902
#130728942
#130719392
#130712142
C22×C11×#1232
#3672011